When dealing with tropical equivalents of coordinate hyperspaces, projective space, toric varieties and other such good things,
one will be confronted with points that have infinite coordinates, undetermined function values and other bad stuff.

Is it worth it? It's totally worth it. We will do intersection theory on tropical toric varieties, with boundary divisors, quotients of homogeneous polynomials, Chow groups and all the good things one expects from the classical theory (and that were absent from tropical intersection theory in R^n, i.e. the talks Tropical Intersection Theory I & II ).